Inference

Two-graph hypothesis testing

graspologic.inference.latent_position_test(A1, A2, embedding='ase', n_components=None, test_case='rotation', n_bootstraps=500)[source]

Two-sample hypothesis test for the problem of determining whether two random dot product graphs have the same latent positions.

This test assumes that the two input graphs are vertex aligned, that is, there is a known mapping between vertices in the two graphs and the input graphs have their vertices sorted in the same order. Currently, the function only supports undirected graphs.

Read more in the tutorials

Parameters:

A1, A2 : nx.Graph, nx.DiGraph, nx.MultiDiGraph, nx.MultiGraph, np.ndarray

The two graphs to run a hypothesis test on. If np.ndarray, shape must be (n_vertices, n_vertices) for both graphs, where n_vertices is the same for both

embedding : string, { 'ase' (default), 'omnibus'}

String describing the embedding method to use:

  • 'ase'
    Embed each graph separately using adjacency spectral embedding and use Procrustes to align the embeddings.
  • 'omnibus'
    Embed all graphs simultaneously using omnibus embedding.

n_components : None (default), or int

Number of embedding dimensions. If None, the optimal embedding dimensions are found by the Zhu and Godsi algorithm.

test_case : string, {'rotation' (default), 'scalar-rotation', 'diagonal-rotation'}

describes the exact form of the hypothesis to test when using 'ase' or 'lse' as an embedding method. Ignored if using 'omnibus'. Given two latent positions, \(X_1\) and \(X_2\), and an orthogonal rotation matrix \(R\) that minimizes \(||X_1 - X_2 R||_F\):

  • 'rotation'
    \[H_o: X_1 = X_2 R\]
  • 'scalar-rotation'
    \[H_o: X_1 = c X_2 R\]

    where \(c\) is a scalar, \(c > 0\)

  • 'diagonal-rotation'
    \[H_o: X_1 = D X_2 R\]

    where \(D\) is an arbitrary diagonal matrix

n_bootstraps : int, optional (default 500)

Number of bootstrap simulations to run to generate the null distribution

Returns:

p_value : float

The overall p value from the test; this is the max of 'p_value_1' and 'p_value_2'

sample_T_statistic : float

The observed difference between the embedded positions of the two input graphs after an alignment (the type of alignment depends on test_case)

misc_stats : dictionary

A collection of other statistics obtained from the latent position test

  • 'p_value_1', 'p_value_2' : float
    The p value estimate from the null distributions from sample 1 and sample 2
  • 'null_distribution_1', 'null_distribution_2' : np.ndarray (n_bootstraps,)
    The distribution of T statistics generated under the null, using the first and and second input graph, respectively. The latent positions of each sample graph are used independently to sample random dot product graphs, so two null distributions are generated

References

[1]Tang, M., A. Athreya, D. Sussman, V. Lyzinski, Y. Park, Priebe, C.E. "A Semiparametric Two-Sample Hypothesis Testing Problem for Random Graphs" Journal of Computational and Graphical Statistics, Vol. 26(2), 2017
graspologic.inference.latent_distribution_test(A1, A2, test='dcorr', metric='euclidean', n_components=None, n_bootstraps=500, workers=1, size_correction=True, pooled=False, align_type='sign_flips', align_kws={}, input_graph=True)[source]

Two-sample hypothesis test for the problem of determining whether two random dot product graphs have the same distributions of latent positions.

This test can operate on two graphs where there is no known matching between the vertices of the two graphs, or even when the number of vertices is different. Currently, testing is only supported for undirected graphs.

Read more in the tutorials

Parameters:

A1, A2 : variable (see description of 'input_graph')

The two graphs, or their embeddings to run a hypothesis test on. Expected variable type and shape depends on input_graph attribute

test : str (default="hsic")

Backend hypothesis test to use, one of ["cca", "dcorr", "hhg", "rv", "hsic", "mgc"]. These tests are typically used for independence testing, but here they are used for a two-sample hypothesis test on the latent positions of two graphs. See hyppo.ksample.KSample for more information.

metric : str or function (default="gaussian")

Distance or a kernel metric to use, either a callable or a valid string. If a callable, then it should behave similarly to either sklearn.metrics.pairwise_distances() or to sklearn.metrics.pairwise.pairwise_kernels(). If a string, then it should be either one of the keys in sklearn.metrics.pairwise.PAIRED_DISTANCES one of the keys in sklearn.metrics.pairwise.PAIRWISE_KERNEL_FUNCTIONS, or "gaussian", which will use a gaussian kernel with an adaptively selected bandwidth. It is recommended to use kernels (e.g. "gaussian") with kernel-based hsic test and distances (e.g. "euclidean") with all other tests.

n_components : int or None (default=None)

Number of embedding dimensions. If None, the optimal embedding dimensions are found by the Zhu and Godsi algorithm. See selectSVD() for more information. This argument is ignored if input_graph is False.

n_bootstraps : int (default=200)

Number of bootstrap iterations for the backend hypothesis test. See hyppo.ksample.KSample for more information.

workers : int (default=1)

Number of workers to use. If more than 1, parallelizes the code. Supply -1 to use all cores available to the Process.

size_correction : bool (default=True)

Ignored when the two graphs have the same number of vertices. The test degrades in validity as the number of vertices of the two graphs diverge from each other, unless a correction is performed.

  • True
    Whenever the two graphs have different numbers of vertices, estimates the plug-in estimator for the variance and uses it to correct the embedding of the larger graph.
  • False
    Does not perform any modifications (not recommended).

pooled : bool (default=False)

Ignored whenever the two graphs have the same number of vertices or size_correction is set to False. In order to correct the adjacency spectral embedding used in the test, it is needed to estimate the variance for each of the latent position estimates in the larger graph, which requires to compute different sample moments. These moments can be computed either over the larger graph (False), or over both graphs (True). Setting it to True should not affect the behavior of the test under the null hypothesis, but it is not clear whether it has more power or less power under which alternatives. Generally not recomended, as it is untested and included for experimental purposes.

align_type : str, {'sign_flips' (default), 'seedless_procrustes'} or None

Random dot product graphs have an inherent non-identifiability, associated with their latent positions. Thus, two embeddings of different graphs may not be orthogonally aligned. Without this accounted for, two embeddings of different graphs may appear different, even if the distributions of the true latent positions are the same. There are several options in terms of how this can be addresssed:

  • 'sign_flips'
    A simple heuristic that flips the signs of one of the embeddings, if the medians of the two embeddings in that dimension differ from each other. See graspologic.align.SignFlips for more information on this procedure. In the limit, this is guaranteed to lead to a valid test, as long as matrix \(X^T X\), where \(X\) is the latent positions does not have repeated non-zero eigenvalues. This may, however, result in an invalid test in the finite sample case if the some eigenvalues are same or close.
  • 'seedless_procrustes'
    An algorithm that learns an orthogonal alignment matrix. This procedure is slower than sign flips, but is guaranteed to yield a valid test in the limit, and also makes the test more valid in some finite sample cases, in which the eigenvalues are very close to each other. See graspologic.align.SignFlips for more information on the procedure.
  • None
    Do not use any alignment technique. This is strongly not recommended, as it may often result in a test that is not valid.

align_kws : dict

Keyword arguments for the aligner of choice, either graspologic.align.SignFlips or graspologic.align.SeedlessProcrustes, depending on the align_type. See respective classes for more information.

input_graph : bool (default=True)

Flag whether to expect two full graphs, or the embeddings.

  • True
    This function expects graphs, either as NetworkX graph objects or as adjacency matrices, provided as ndarrays of size (n, n) and (m, m). They will be embedded using adjacency spectral embeddings.
  • False
    This function expects adjacency spectral embeddings of the graphs, they must be ndarrays of size (n, d) and (m, d), where d must be same. n_components attribute is ignored in this case.
Returns:

p_value : float

The overall p value from the test.

sample_T_statistic : float

The observed difference between the embedded latent positions of the two input graphs.

misc_stats : dictionary

A collection of other statistics obtained from the latent position test

  • null_distribution : ndarray, shape (n_bootstraps,)
    The distribution of T statistics generated under the null.
  • n_components : int
    Number of embedding dimensions.
  • Q : array, size (d, d)
    Final orthogonal matrix, used to modify X.

References

[1]Tang, M., Athreya, A., Sussman, D. L., Lyzinski, V., & Priebe, C. E. (2017). "A nonparametric two-sample hypothesis testing problem for random graphs." Bernoulli, 23(3), 1599-1630.
[2]Panda, S., Palaniappan, S., Xiong, J., Bridgeford, E., Mehta, R., Shen, C., & Vogelstein, J. (2019). "hyppo: A Comprehensive Multivariate Hypothesis Testing Python Package." arXiv:1907.02088.
[3]Alyakin, A. A., Agterberg, J., Helm, H. S., Priebe, C. E. (2020). "Correcting a Nonparametric Two-sample Graph Hypothesis Test for Graphs with Different Numbers of Vertices" arXiv:2008.09434